### Mathematics and sex

In reference to Curt’s latest post, I feel obligated to make a quick comment on the linked discussion between Steven Pinker and Elizabeth Spelke, specifically Spelke’s comments on mathematical aptitude. Now, I’m not (yet) a card-carrying member of the mathematical establishment, but I like to think two years in an Ivy League Ph.D. program has given me some insight into how mathematics works, certainly more than I think Spelke, despite her apparently solid reputation as a psychologist, demonstrates. Her argument is that any difference between men and women in performance in the hard sciences and mathematics is socially, not biologically, determined.^{1} Okay, as valid a hypothesis as any other. What’s her evidence?

Well, it turns out to be surprisingly elusive, especially coming on the heels of Pinker’s well-reasoned argument that there is some biological basis for the difference of performance between men and women. Of course, she’s right that it’s misleading to look at, e.g., the SAT math test to provide a definitive answer, because the writers of that test can tweak it to get pretty much any result they want (although it *is* interesting to note that she apparently doesn’t even consider the possibility that the writers of the test are trying to create a test that most closely tests mathematical aptitude, whatever that is, instead talking about how “they can create a test that makes women look like better mathematicians, or a test that makes men look like better mathematicians”). That having been said, it seems downright disingenuous to me for her to *acknowledge* that males and females tend to have different cognitive profiles, while denying that there’s any chance that has an effect on aptitude for mathematics:^{2}

Finally, the mathematical word problems on the SAT-M very often allow multiple solutions. Both item analyses and studies of high school students engaged in the act of solving such problems suggest that when students have the choice of solving a problem by plugging in a formula or by doing Ven [

sic] diagram-like spatial reasoning, girls tend to do the first and boys tend to do the second.

This comes as a continual surprise to non-mathematicians (who imagine that mathematicians sit around doing more and more complicated arithmetic and calculus problems all day), but plugging into a formula is *virtually worthless* from a mathematical perspective, whereas “Ven[n] diagram-like spatial reasoning” is fundamentally similar to the sort of thinking that a professional mathematician does. Thus, if women tend to be plug-and-chug types, it shouldn’t really be a surprise that they are underrepresented in mathematics departments. Of course, this doesn’t demonstrate that there’s any biological basis to the difference, but Spelke’s apparent contention that plug-and-chug methodology and more abstract reasoning constitute equivalent levels of mathematical aptitude seems pretty naïve.

That having been said, she does make a strong argument when she points out that women and men get equal grades in math classes in college and are math majors in roughly equal numbers. However, it needs to be pointed out that undergraduate math courses and professional mathematics are *qualitatively* different, not just quantitatively, which Spelke implicitly assumes:

I suggest the following experiment. We should take a large number of male students and a large number of female students who have equal educational backgrounds, and present them with the kinds of tasks that real mathematicians face. We should give them new mathematical material that they have not yet mastered, and allow them to learn it over an extended period of time: the kind of time scale that real mathematicians work on. We should ask, how well do the students master this material? The good news is, this experiment is done all the time. It’s called high school and college.

The qualitative difference is the following: in undergraduate math courses (at least in my experience), performance is based largely on one’s ability to internalize a few examples and follow their template in solving other (relatively easy) problems; the professional mathematician must take known results and integrate them in a novel way to solve problems nobody has ever solved before (which, given that they are unsolved, are pretty much universally very, very difficult). The former is, needless to say, much more amenable to the plug-and-chug mindset than the latter.

Spelke summarizes this section of her argument as follows:

The outcome of this large-scale experiment gives us every reason to conclude that men and women have equal talent for mathematics. Here, I too would like to quote Diane Halpern. Halpern reviews much evidence for sex differences, but she concludes, “differences are not deficiencies.” Men and women have equal aptitude for mathematics. Yes, there are sex differences, but they don’t add up to an overall advantage for one sex over the other.

Again, this is just disingenuous. The outcome of this large-scale experiment is not that men and women have equal talent for “mathematics”; it is that they have equal talent for undergraduate mathematics classes. Certainly, high performance in undergraduate math classes is a prerequisite for getting into graduate school, which is, in turn, a prerequisite for getting a Ph.D. and becoming a math professor, but, as untold grad school dropouts can tell you, there’s a hell of a difference between the sort of thinking that you do as an undergrad and the sort of thinking you must do as a “real” mathematician (in this context, it’s telling that Spelke uses the fact that 57% of accountants are women as evidence that women have the same mathematical aptitude as men). Of course, maybe it shouldn’t be a surprise that a psychologist has an apparently naïve view of what constitutes professional-grade mathematical aptitude when psychology styles itself (these days, anyway) as an empirical science, which is to say an analytic discipline, while mathematics is practically the definition of a synthetic discipline.

Now, this is not to say that there aren’t significant social causes of the male/female discrepancy in the hard sciences and mathematics (in fact, I really haven’t said anything at all about biology; it’s certainly possible that *all* of the above differences are due to social factors). Spelke makes good points about how parents seem to perceive the performance and capabilities of male vs. female children differently and how faculty hiring committees tend to receive male candidates more favorably than equally qualified female candidates (this latter should come as no surprise to readers of Malcolm Gladwell’s *Blink*, which touches on the fact that a significant contributor to the increasing gender balance of classical symphonies in the last few decades is the fact that virtually all reputable symphonies these days conduct auditions with the candidate performing behind a screen). These and probably many other social factors almost certainly play a role in women’s under-representation in math and science; as may be, Spelke’s apparent ignorance of mathematics makes it hard to accept her position on the issue which is, as Pinker rightly points out at the beginning of his presentation, extreme.

1. I’m quite aware that in this post I’m cherry-picking from Spelke’s argument by addressing only that component of it which I feel like I have some expertise in. I’m not trying to offer a comprehensive rebuttal of her argument and just because I disagree with what is essentially one point in a larger argument do I mean to suggest that the other points are also wrong. As usual, it’s probably best to read it for yourself and draw your own conclusions.

2. The usual caveat applies: when I talk about men or women having more or less aptitude for something, I’m speaking of statistical averages (to whatever degree those even make sense), not of individual people. There are plenty of women who are wonderful mathematicians (some of whom I’m lucky enough to know) and countless men who are abject morons (many of whom I also know); the old adage that statistics lie and liars use statistics is never more true than when someone tries to use statistics to “prove” statements about individuals.

May 19th, 2005 at 5:20 am

I’m not sure of the true importance of this point, but I find it interesting that Spelke (and Pinker, to a lesser extent), pretty much delineate

scientificability down to ability in mathematics. You’ve picked up on the issue of relative mathematical ability between the sexes, but of course we should keep in mind that both of them are ostensibly talking about scientific ability, and it seems to me that the presumption that the two are identical (which of course is their presumption, not yours) is somewhat unjustified. It’s true that mathematical ability is essential to science, but I recall David Hilbert saying about Einstein something to the effect of: “Every schoolboy in Göttingen knows more about the equations than him, but he did the [conceptual] work.” His doctoral advisor habitually referred to him as a “lazy dog” for working out problems representationally rather than through equations. That is perhaps the most extreme example, but it seems to me that many of the sciences, particularly physics, are more about spatial representation and conceptual modelling even than front-line mathematics, and less about analytical equation-solving.p.s. I’d say Spelke is more naïve than disingenuous for thinking that because male and female babies are equally good at counting objects they will later be equally talented astrophysicists, but I would say that her main disingenuity is by following back on the old canard, which so few on her side of the debate seem to be able to resist, that even if women are temperamentally less suited to professional science than men, that probably means that the whole system of science is unfairly biased, i.e. too competition-driven, not compassionate enough, etc. This is, to put it mildly, a bit of a departure from the premises of the debate. The claim that if current science practice does favor men then we are somehow not “doing” science right just shows how pre-determined and inflexible the agenda is, which is not turning out the most highly-qualified scientists. Plus, if you throw out competition, or rather the objective criteria by which competition becomes meaningful, you have social science, in which women are indeed much more prominent, but I don’t know of too many who would brag of the achievements of the social sciences in the last hundred years relative to the physical sciences, and plus, based on my experience, in these disciplines the environment is actually a lot less compassionate and more vicious because, in the absence of objective standards of accomplishment, those involved seem to be aware that the only way they can truly distinguish themselves from their colleagues is by demolishing their reputations.

May 19th, 2005 at 12:15 pm

You’ve picked up on the issue of relative mathematical ability between the sexes, but of course we should keep in mind that both of them are ostensibly talking about scientific ability, and it seems to me that the presumption that the two are identical (which of course is their presumption, not yours) is somewhat unjustified.I definitely agree with this, although maybe not precisely in the sense you mean. One could make an argument that, except in the furthest reaches of theoretical physics, scientists use math much more in the “plug-and-chug” style than in the “conceptual” style. However, scientists, presumably (I’ve never been much good at science myself, so I can’t say definitively) must be more conceptual within the context of their own subject matter (this is certainly the case in theoretical physics, as Hilbert’s comment on Einstein illustrates, but I assume something similar holds for the more “prosaic” sciences as well).

Of course, one must keep in mind that mathematicians tend to have a bit of an institutional disdain for scientists as mere number-crunchers, so that perspective may be coloring my comments.